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Volume 2013 (2013), Article ID 106393, 7 pages
Lagrange Spaces with -Metric
Department of Mathematics, University of Allahabad, Allahabad 211002, India
Received 31 October 2012; Accepted 9 January 2013
Academic Editor: Matthew He
Copyright © 2013 Suresh K. Shukla and P. N. Pandey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study Lagrange spaces with -metric, where is a cubic metric and is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a -metric. Several other properties of such space are also discussed.
Finsler spaces endowed with -metric were studied by several geometers such as Matsumoto [1, 2] and Kitayama et al. , and various important applications of such spaces have been observed in physics and relativity theory (cf. [4, 5]). The notion of -metric was taken to a more general space called Lagrange space and the study was performed by the authors such as Miron , Nicolaescu [7, 8], and the present authors . An -dimensional Lagrange space is said to be endowed with -metric if Lagrangian is function of and only, that is, being a Riemannian metric and a 1-form. Recently, Pandey and Chaubey  discussed Lagrange spaces with -metric and obtained several results. They called a Lagrange space to be endowed with -metric if Lagrangian is a function of and only, that is, where is a cubic metric and is a 1-form, that is, and . The paper  by Pandey and Chaubey is full of flaws and needs to be rectified. The aim of the present paper is to develop a revised and modified theory of Lagrange spaces with -metric.
The paper is organized as follows. In Section 2, we define a Lagrange space and discuss some preliminary results required for the discussion of the following sections. It includes the notion of a Lagrange space with -metric. In Section 3, we discuss some properties of a Lagrange space with -metric and obtain the expression for the fundamental metric tensor and its inverse . In Section 4, we consider the variational problem in Lagrange spaces with -metric and obtain various forms of Euler-Lagrange equations. Section 5 deals with the semispray of a Lagrange space with -metric. In Section 6, we obtain the coefficients of nonlinear connection in a Lagrange space endowed with -metric. Section 7 consists of concluding remarks on the results obtained in the paper.
Let be an -dimensional smooth manifold and let be its tangent bundle. Let and be local coordinates on and , respectively. A Lagrangian is a function which is a smooth function on and continuous on the null section. The Lagrangian is said to be regular if rank , where is a covariant symmetric tensor called the fundamental metric tensor of the Lagrangian . A Lagrange space is a pair being a regular Lagrangian whose metric tensor has constant signature on .
The integral of action of the Lagrangian along a smooth curve leads to the following Euler-Lagrange equations:
The coefficients of the semispray of a Lagrange space are given by The semispray is called a canonical semispray as its coefficients depend on only.
The coefficients of canonical nonlinear connection of a Lagrange space are given by
In the present paper, we study a Lagrange space whose Lagrangian is a function of and only, where Let us denote this Lagrangian by . Thus
The space is called a Lagrange space with -metric (cf. ). Following are some examples of regular Lagrangians with -metric: The Lagrange space determined by the Lagrangian in (8)(iii) is reducible to a Finsler space whereas those determined by the Lagrangians in (8)(i) and (8)(ii) are not so.
3. Fundamental Metric Tensor of
If we differentiate (5) partially with respect to and use the symmetry of in its indices, we obtain where .
Again differentiating (9) partially with respect to , using symmetry of in its indices and simplifying, we find where .
Proposition 1. In a Lagrange space with -metric, the following hold good:
The moments of Lagrangian are given by In our case, the Lagrangian is a function of and only (vide (7)). Therefore, we have where .
Theorem 2. In a Lagrange space with -metric, the moments of Lagrangian are given by where
Remarks 3. The scalars and appearing in Theorem 2 are called the principal invariants of the space .
Differentiating (19) and (20), partially with respect to and simplifying, we, respectively, have where Thus, we have the following.
Proposition 4. The derivatives of the principal invariants of a Lagrange space with -metric are given by with
The energy of Lagrangian is defined as Using (7) in (26), we have Since and are positively homogeneous of degree one in , by virtue of Euler’s theorem on homogeneous functions, we have In view of (28), (27) takes the form Thus, we have the following.
Theorem 5. In a Lagrange space with -metric, the energy of the Lagrangian is given by (29).
Now, we find out expression for the fundamental metric tensor of a Lagrange space with -metric. Using (7) in (1), we have In view of Proposition 1, (30) takes the form Equation (31) can be written as where and satisfy Thus, we have the following.
Theorem 6. The fundamental metric tensor of a Lagrange space with -metric is given by (32).
The following result gives the expression for the inverse of .
Theorem 7. The inverse of the fundamental metric tensor of a Lagrange space with -metric is given by where
Remarks 8. In , Pandey and Chaubey obtained the following:
where and are, respectively, given by (19) and (20),
where and is the same as given in (25),
where and satisfy , and
where and .
In view of corresponding results obtained by us, these results are erroneous.
4. Euler-Lagrange Equations
Theorem 9. The Euler-Lagrange equations of a Lagrange space with -metric are of the following form:
For the natural parametrization of the curve with respect to the cubic metric . Thus, we have the following.
Theorem 10. In the natural parametrization, the Euler-Lagrange equations of a Lagrange space with -metric are
If is constant on the integral curve of the Euler-Lagrange equations with natural parametrization, then (52) takes the form Thus, we have the following.
Theorem 11. If is constant along the integral curve of the Euler-Lagrange equations with natural parametrization, then the Euler-Lagrange equations of the Lagrange space with -metric are given by (53).
Remarks 12. Pandey and Chaubey  obtained the following form of Euler-Lagrange equations:
In case of natural parametrization, their result is
If is constant along the integral curve of the Euler-Lagrange equations (with natural parametrization), the Euler-Lagrange equations obtained by Pandey and Chaubey  are given by
In view of Theorem 9, Theorem 10, and Theorem 11, all the above discussed results of Pandey and Chaubey  are erroneous.
5. Canonical Semispray
In this section, we obtain the coefficients of the canonical semispray of a Lagrange space with -metric.
Using (7) in (3), we obtain Since and , we have where Using (58), (19), and (20) in , we get Differentiating (60) partially with respect to and simplifying, we have where Using (60) and (61) in (57), we obtain where Thus, we have the following.
Theorem 13. The local coefficients of canonical semispray of a Lagrange space with -metric are given by (63).
6. Canonical Nonlinear Connection
In this section, we obtain the local coefficients of the canonical nonlinear connection of a Lagrange space with -metric.
Partial differentiation of , with respect to , yields If we partially differentiate the quantities appearing in (25) and (64) with respect to , we find the following quantities: where and denotes interchanging indices and and taking sum. Also, we have Now, applying (63) in (4), we get Using (24), (62)–(64), (65), (66) and (68) in (69) and simplifying, we obtain where .
Thus, we have the following.
Theorem 14. The local coefficients of the canonical nonlinear connection of a Lagrange space with -metric are given by (70).
We have developed the theory of Lagrange spaces with -metric. The paper presents a significant generalization of the theory of Lagrange spaces with -metrics (cf. [6–8]). The expressions for the geometric objects obtained in the paper may be useful in further work on the spaces under consideration. The importance of the results lies in the study of canonical metrical -connection, curvatures, and torsions in such spaces. The expressions for canonical semispray and nonlinear connection, obtained, respectively, in Section 5 and Section 6, may be applicable in geodesic correspondences between two Lagrange spaces with different -metrics on the same underlying manifold. It is a matter of later investigations to look into the aforesaid applications of the results obtained in the paper.
The authors are thankful to the reviewer for his/her valuable comments and suggestions. S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.
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